Within a Bayesian formulation of the inverse problem, associated with the mapping (REF ) and measurements (REF ), one encodes statistical information about the system (say inputs \(u\)  in (REF )) in terms of a prior probability measure. The additional information from the measurements (REF ) can be used to improve the prior by an application of the well-known Bayes' theorem [1]}. This results in a so-called posterior probability measure, on the inputs \(u\) , which represents the conditional probability of the underlying inputs, given the measurements (REF ). Thus, the Bayesian Inverse Problem can be interpreted as a mapping from the measurements (REF ) to the posterior measure.
